Odd that there is not a mic sensitivity adjustment on an audio panel...there is most likely a attenuation potentiometer somewhere, check the manual...but on the Zulus there is a small plastic dot cap on the boom down by the mic...pop that off and with a micro screwdriver you can adjust the mic gain on the headset itself.
Turning the gain down is essentially turning the volume of the mic down by sending less signal to the panel that triggers the gate to open.
The "gate" or "gating" is what triggers the mic to be automatically open or closed in the audio system. It hears sound, it opens the gate (turns the mic on)...no sound, gate closes the gate so you don't hear background noise. That threshold of how much level it takes to open that gate is what is usually adjustable on the audio panel. A gate threshold is rarely an unadjustable fixed setting even in older equipment, there are too may variables. Your gate threshold is set to low, so your breathing sound is triggering the gate to open rather then being triggered by speech. Reducing the mic gain should solve that issue if you can not solve it on the audio panel side but that is also going to make your headset mic broadcast quieter in the rest of the system to other PAX.
While microphone "squelch" is technically the correct term...on aviation audio panels the "squelch" also typically refers to the gate threshold setting for receiving radio transmissions over the air so don't get that term confused when looking it up in the manual.